Final exam formula sheet - Disk Method =]2 Washer Method =]2]2 2 Shell Method = 2 2 Arc Length Formula = 1 Useful Trigonometric Identities cos 2 =

Final exam formula sheet - Disk Method =]2 Washer Method...

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Disk Method : 𝑉 = ∫ 𝜋 [ 𝑅 ( 𝑥 )] 2 𝑑𝑥 𝑏 𝑎 Washer Method : 𝑉 = ∫ 𝜋 ([ 𝑅 ( 𝑥 )] 2 [ 𝑟 ( 𝑥 )] 2 ) 𝑑𝑥 𝑏 𝑎 Shell Method : 𝑉 = 2 𝜋𝑟ℎ 𝑑𝑥 𝑏 𝑎 Surface Area : 𝑆 = 2 𝜋 ∙ 𝑓 ( 𝑥 ) 1 + �𝑓 ( 𝑥 ) 2 𝑑𝑥 𝑏 𝑎 Arc Length Formula : 𝐿 = 1 + �𝑓 ( 𝑥 ) 2 𝑑𝑥 𝑏 𝑎 Useful Trigonometric Identities : cos 2 𝜃 = 1+cos 2𝜃 2 ; sin 2 𝜃 = 1−cos 2𝜃 2 ; sin 2 𝜃 = 2 sin 𝜃 cos 𝜃 The 𝒏 -th Term Test for Divergence : 𝑎 𝑛 𝑛=1 diverges if lim 𝑛→∞ 𝑎 𝑛 fails to exist or is different from zero. The Limit Comparison Test : Let ∑ 𝑎 𝑛 and ∑ 𝑏 𝑛 be series with positive terms and suppose 𝜌 = lim 𝑛→∞ 𝑎 𝑛 𝑏 𝑛 . a) If 𝜌 is finite and 𝜌 > 0 , then the series both converge or both diverge. b) If 𝜌 = 0 and ∑ 𝑏 𝑛 converges, then ∑ 𝑎 𝑛 converges. c) If 𝜌 = and ∑ 𝑏 𝑛 diverges, then ∑ 𝑎 𝑛 diverges.

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